Hilbert transforms. Volume 2 / Frederick W. King.

Format:

Book

Author/Creator:

King, Frederick W., 1947- author.

Series:

Encyclopedia of mathematics and its applications ; v. 125.

Encyclopedia of mathematics and its applications ; volume 125

Language:

English

Subjects (All):

Hilbert transform.

Physical Description:

1 online resource (xxxviii, 660 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2009.

Language Note:

English

Summary:

The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wide-ranging collection of references both to the classical mathematical papers, and to a diverse array of applications.

Contents:

Cover; Title; Copyright; Dedication; Contents; Preface; List of symbols; List of abbreviations; 15 Hilbert transforms in En; 15.1 Definition of the Hilbert transform in En; 15.2 Definition of the n-dimensional Hilbert transform; 15.3 The double Hilbert transform; 15.4 Inversion property for the n-dimensional Hilbert transform; 15.5 Derivative of the n-dimensional Hilbert transform; 15.6 Fourier transform of the n-dimensional Hilbert transform; 15.7 Relationship between the n-dimensional Hilbert transform and translation and dilation operators; 15.8 The Parseval-type formula

15.9 Eigenvalues and eigenfunctions of the n-dimensionalHilbert transform15.10 Periodic functions; 15.11 A Calderón

Zygmund inequality; 15.12 The Riesz transform; 15.13 The n-dimensional Hilbert transform of distributions; 15.14 Connection with analytic functions; Notes; Exercises; 16 Some further extensions of the classical Hilbert transform; 16.1 Introduction; 16.2 An extension due to Redheffer; 16.3 Kober's definition for the L case; 16.4 The Boas transform; 16.4.1 Connection with the Hilbert transform; 16.4.2 Parseval-type formula for the Boas transform

16.4.3 Iteration formula for the Boas transform16.4.4 Riesz-type bound for the Boas transform; 16.4.5 Fourier transform of the Boas transform; 16.4.6 Two theorems due to Boas; 16.4.7 Inversion of the Boas transform; 16.4.8 Generalization of the Boas transform; 16.5 The bilinear Hilbert transform; 16.6 The vectorial Hilbert transform; 16.7 The directional Hilbert transform; 16.8 Hilbert transforms along curves; 16.9 The ergodic Hilbert transform; 16.10 The helical Hilbert transform; 16.11 Some miscellaneous extensions of the Hilbert transform; Notes; Exercises; 17 Linear systems and causality

17.1 Systems17.2 Linear systems; 17.3 Sequential systems; 17.4 Stationary systems; 17.5 Primitive statement of causality; 17.6 The frequency domain; 17.7 Connection to analyticity; 17.7.1 A generalized response function; 17.8 Application of a theorem due to Titchmarsh; 17.9 An acausal example; 17.10 The Paley

Wiener log-integral theorem; 17.11 Extensions of the causality concept; 17.12 Basic quantum scattering: causality conditions; 17.13 Extension of Titchmarsh's theorem for distributions; Notes; Exercises; 18 The Hilbert transform of waveforms and signal processing

18.1 Introductory ideas on signal processing18.2 The Hilbert filter; 18.3 The auto-convolution, cross-correlation, and auto-correlation functions; 18.4 The analytic signal; 18.5 Amplitude modulation; 18.6 The frequency domain; 18.7 Some useful step and pulse functions; 18.7.1 The Heaviside function; 18.7.2 The signum function; 18.7.3 The rectangular pulse function; 18.7.4 The triangular pulse function; 18.7.5 The sinc pulse function; 18.8 The Hilbert transform of step functions and pulse forms; 18.9 The fractional Hilbert transform: theLohmann

Mendlovic

Zalevsky definition

18.10 The fractional Fourier transform

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

ISBN:

0-511-96510-9

1-107-10319-3

1-107-09504-2

0-511-73527-8

1-107-08888-7

1-107-09177-2

OCLC:

862939808